Katznelson's recurrence conjecture for countably infinite abelian groups

Let $G$ be a countably infinite abelian group. A set $S\subseteq G$ is a set of Bohr recurrence if it intersects every Bohr neighborhood of $0$, and it is a set of chromatic recurrence if the Cayley graph determined by $S$ has infinite chromatic number.

Katznelson's conjecture. If $S\subseteq G$ is a set of Bohr recurrence, then $S$ is a set of chromatic recurrence.

This is the general form of a question suggested by Veech and popularized by Katznelson. No countably infinite abelian group is currently known for which the answer is settled.